No real "true charger" definition objectively.
The most important element afaic is current-limiting. Lower-cost latching or even hiccup mode OCP interrupts the charging process, you really want the ability to transition to constant current mode,
since LI chemistries will always demand a higher amps load than what's good for them, and/or will release the magic smoke on unprotected sources.
This applies to PSUs as well as "chargers"
The key in that distinction afaic is an automated stop-charge function. Some claim to be chargers but don't even do that.
Ideally can be adjusted by the user, sometimes just a simple egg-timer, sometimes an algorithm that tries to adjust automatically, the ideal is based on tapering amps measured at the pack.
The Bulk/CC vs Absorb/CV stage transition you describe is not controlled by anything external, not by a human nor the charge source regulator, but
inherent in and controlled by the battery, varies by C-rate, SoC, chemistry resistance/CAR etc.
Some "stage" chargers drop from Absorb voltage down to a lower Float after the charge process is finished, but that is not so needed in this context.
But **adjustable** voltage setpoints, ideally user custom variable within a wide range to me is critical.
And an adjustable output current is IMO also important, even if only a few canned choices.
These last two are more commonly available in "lab style" PSUs, units marketed as "chargers" with these abilities are generally expensive.
Note all the above only applies to the unit actually connected to the battery. If you go with a hobby charger powered by a PSU, the latter can be dumb as rocks, all the smarts are in the DC-DC charger.
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So, current limiting is essential, really a CC mode. If not adjustable then the limited output will safely charge only a certain Ah range to keep to an appropriate C-rate.
If there is no stop-charge intelligence, then you the owner needs to monitor the V/A output and pull the plug when the time comes.
Depending on the condition of the batteries, relative V / A of the source, doing any of this wrong can result in "unexpected thermal runaway events", boom bad.
So never mess around inside your living space, do it outdoors or in a back shed, maybe a suitable garage.
And never leave charging unattended.
Unless you spend a lot more money on infrastructure and go to a safer chemistry like LFP or maybe LTO, but sounds like not in your situation.